MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

• Published in: Computational optimization and applications Vol. 70; no. 1; pp. 267 - 294
• Main Authors: Gugat, Martin, Leugering, Günter, Martin, Alexander, Schmidt, Martin, Sirvent, Mathias, Wintergerst, David
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2018; Springer Nature B.V
• Subjects: Computational fluid dynamics; Convex and Discrete Geometry; Euler-Lagrange equation; Gas transport; Management Science; Mathematics; Mathematics and Statistics; Nonlinear control; Nonlinear equations; Operations Research; Operations Research/Decision Theory; Optimization; Ordinary differential equations; Partial differential equations; Solvers; Stability; Statistics; Switching theory; Transportation networks; Well posed problems; 76B75; Mixed-integer optimal control; 49J20; 90C35; Mixed-integer linear optimization; 90C11; Instantaneous control; Partial differential equations on graphs; Gas networks; 49J15
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-017-9970-1

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.